In a `DeltaABC, angleB=90^(@), angleC=45^(@)` and D is mid point of AC. If AC = `4sqrt(2)` unit then BD is

Options

A. `5/2` unit
B. 2 unit
C. `2sqrt(2)` unit
D. `4sqrt(2)` unit

Answer

To solve the problem step by step, we will analyze the triangle ABC with the given angles and the length of side AC.

### Step 1: Identify the Triangle Properties
We have triangle ABC where:
- Angle B = 90°
- Angle C = 45°
- Therefore, angle A = 180° - 90° - 45° = 45°.

This means triangle ABC is an isosceles right triangle (45°-45°-90° triangle).

**Hint:** Remember that in a triangle, the sum of angles is always 180°.

### Step 2: Determine the Lengths of the Sides
Since AC = 4√2 units and D is the midpoint of AC, we can find the lengths of AD and DC:
- AD = DC = AC / 2 = (4√2) / 2 = 2√2 units.

**Hint:** The midpoint divides a segment into two equal parts.

### Step 3: Use Pythagorean Theorem to Find AB and BC
In triangle ABC, since it is an isosceles right triangle, we can denote the lengths of AB and BC as 'x'. According to the Pythagorean theorem:
- AC² = AB² + BC²
- (4√2)² = x² + x²
- 32 = 2x²
- x² = 16
- x = 4 units.

Thus, AB = 4 units and BC = 4 units.

**Hint:** In a 45°-45°-90° triangle, the legs are equal, and the hypotenuse is √2 times the length of each leg.

### Step 4: Calculate BD
Now, we need to find the length of BD. We can use the Pythagorean theorem in triangle BDC:
- BD² = BC² + DC²
- BD² = 4² + (2√2)²
- BD² = 16 + 8
- BD² = 24
- BD = √24 = 2√6 units.

**Hint:** When applying the Pythagorean theorem, ensure you correctly identify the lengths of the sides involved.

### Final Answer
Thus, the length of BD is **2√6 units**.

**Summary of Steps:**
1. Identify triangle properties and angles.
2. Find lengths of sides using the midpoint.
3. Use the Pythagorean theorem to find the lengths of the other sides.
4. Calculate BD using the Pythagorean theorem in triangle BDC.